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12. Rotational Kinematics

Equations of Rotational Motion

Physics

12. Rotational Kinematics

Equations of Rotational Motion

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3:09 minutes

Problem 10a

Textbook Question

An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev/min to 200 rev/min in 4.00 s. Find the angular acceleration in rev/s² and the number of revolutions made by the motor in the 4.00-s interval.

Verified step by step guidance

Step 1: Convert the initial and final angular velocities from revolutions per minute (rev/min) to revolutions per second (rev/s). Use the conversion factor: 1 minute = 60 seconds. For example, \( \omega_{i} = \frac{500}{60} \) rev/s and \( \omega_{f} = \frac{200}{60} \) rev/s.

Step 2: Calculate the angular acceleration \( \alpha \) using the formula for uniform angular acceleration: \( \alpha = \frac{\omega_{f} - \omega_{i}}{\Delta t} \), where \( \Delta t \) is the time interval (4.00 s). Substitute the values of \( \omega_{i} \), \( \omega_{f} \), and \( \Delta t \) to find \( \alpha \) in rev/s².

Step 3: Determine the number of revolutions made by the fan during the 4.00-s interval. Use the kinematic equation for rotational motion: \( \theta = \omega_{i} \Delta t + \frac{1}{2} \alpha \Delta t^2 \), where \( \theta \) is the angular displacement in revolutions. Substitute the values of \( \omega_{i} \), \( \alpha \), and \( \Delta t \) to calculate \( \theta \).

Step 4: To find the additional time required for the fan to come to rest, use the formula \( \Delta t = \frac{\omega_{f} - \omega_{i}}{\alpha} \), where \( \omega_{f} = 0 \) (since the fan comes to rest) and \( \omega_{i} \) is the final angular velocity from part (a). Substitute the values of \( \omega_{i} \) and \( \alpha \) to calculate the additional time.

Step 5: Combine the results from part (a) and part (b) to understand the total behavior of the fan's motion. Ensure all units are consistent and verify the calculations for accuracy.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, typically expressed in revolutions per minute (rev/min) or radians per second. In this problem, the fan's angular velocity decreases from 500 rev/min to 200 rev/min, indicating a change in its rotational speed over time.

Angular Acceleration

Angular acceleration refers to the rate of change of angular velocity over time, usually expressed in rev/s² or rad/s². It can be calculated by taking the difference in angular velocity and dividing it by the time interval during which the change occurs. In this case, it helps determine how quickly the fan slows down.

Kinematics of Rotational Motion

The kinematics of rotational motion involves equations that describe the motion of rotating objects, similar to linear motion but in terms of angular quantities. Key equations relate angular displacement, angular velocity, and angular acceleration, allowing us to calculate the total revolutions made by the fan during its deceleration and the time required to come to a complete stop.

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Related Practice

Multiple Choice

A tiny object spins with 5 rad/s around a circular path of radius 10 m. The object then accelerates at 3 rad/s². Calculate its angular speed 8 s after starting to accelerate.

BONUS:Calculate its linear displacement in the 8 s.

The turntable of a DJ set is spinning at a constant rate just before it is turned off. If the turntable decelerates at 3 rad/s² and goes through an additional 30 rotations before stopping, how fast (in RPM) was the turntable initially spinning?

BONUS:How long (in seconds) does the turntable take to stop?

A high-speed flywheel in a motor is spinning at 500 rpm when a power failure suddenly occurs. The flywheel has mass 40.0 kg and diameter 75.0 cm. The power is off for 30.0 s, and during this time the flywheel slows due to friction in its axle bearings. During the time the power is off, the flywheel makes 200 complete revolutions. At what rate is the flywheel spinning when the power comes back on?

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Textbook Question

An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev/min to 200 rev/min in 4.00 s. How many more seconds are required for the fan to come to rest if the angular acceleration remains constant at the value calculated in part (a)?

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Textbook Question

A wheel is rotating about an axis that is in the z-direction. The angular velocity ω_z is -6.00 rad/s at t = 0, increases linearly with time, and is +4.00 rad/s at t = 7.00 s. We have taken counterclockwise rotation to be positive. What is the angular displacement of the wheel at t = 7.00 s?

916

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Textbook Question

A wheel is rotating about an axis that is in the z-direction. The angular velocity ω_z is -6.00 rad/s at t = 0, increases linearly with time, and is +4.00 rad/s at t = 7.00 s. We have taken counterclockwise rotation to be positive. Is the angular acceleration during this time interval positive or negative?

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Textbook Question

An electric turntable 0.750 m in diameter is rotating about a fixed axis with an initial angular velocity of 0.250 rev/s and a constant angular acceleration of 0.900 rev/s².

(a) Compute the angular velocity of the turntable after 0.200 s.

(b) Through how many revolutions has the turntable spun in this time interval?

936

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Textbook Question

A wheel is rotating about an axis that is in the z-direction. The angular velocity ω_z is -6.00 rad/s at t = 0, increases linearly with time, and is +4.00 rad/s at t = 7.00 s. We have taken counterclockwise rotation to be positive. During what time interval is the speed of the wheel increasing? Decreasing?

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An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev/min to 200 rev/min in 4.00 s. Find the angular acceleration in rev/s2 and the number of revolutions made by the motor in the 4.00-s interval.

Key Concepts

Angular Velocity

Angular Acceleration

Kinematics of Rotational Motion

Watch next

An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev/min to 200 rev/min in 4.00 s. Find the angular acceleration in rev/s² and the number of revolutions made by the motor in the 4.00-s interval.